What is a Fractal?

Date: 04/01/97 at 17:16:47
From: brandin
Subject: Fractals
What are fractals? Who first used the term, when, why, and give at
least two specific examples.

Date: 04/01/97 at 18:29:40
From: Doctor Sarah
Subject: Re: Fractals
Hi Brandin -
There's a lot on the Web about fractals. Much of it is hard to
understand, but there are sites that will give you good explanations
and illustrations. Here is some information compiled from what the
Web has to offer, and you will want to read more yourself at the sites
listed:
On his Web page about fractals,
http://www.glyphs.com/art/fractals/what_is.html
Alan Beck says, "Basically, a fractal is any pattern that reveals
greater complexity as it is enlarged. Thus, fractals graphically
portray the notion of 'worlds within worlds' which has obsessed
Western culture from its tenth-century beginnings."
He further explains that when we look very closely at patterns that
are Euclidean, the shapes look more and more like straight lines, but
that when you look at a fractal up close you see more and more
details.
He illustrates with several graphics. Here is a particularly nice one
(it is also 96K, so be patient and wait for it):
http://www.glyphs.com/art/fractals/images/serpente.jpg
"Whether generated by computers or natural process, all fractals are
spun from what scientists call a 'positive feedback loop'. Something -
data or matter - goes in one 'end', undergoes a given, often very
slight, modification, and comes out the other. Fractals are produced
when theoutput is fed back into the system as input again and again."
Students often study Sierpinski's Triangle as an example of a fractal.
You start with one triangle. [Level Zero.] Then you mark the midpoint
on each of the three sides and draw a line from Midpoint 1 to Midpoint
2 to Midpoint 3. You will have the original larger triangle, and
inside it will be four smaller triangles, three pointing up and one
down. [Level One.]
You can see this process at Cynthia Lanius' Web unit on fractals:
http://math.rice.edu/~lanius/fractals/
Here the triangles have been shaded in, and are black.
Levels One, Two, etc. are called "iterations" - repetitions of the
same process, where the output of one level becomes the input for the
next.
Fractals can be made using different functions. Sierpinski's Triangle
is made by continually dividing a triangle into other triangles, on
and on and on. Koch's snowflake - see Cynthia Lanius' page at
http://math.rice.edu/~lanius/frac/koch.html
- is made by starting with a large equilateral triangle, making a
star, dividing one side of the triangle into three parts, taking out
the middle section and replacing it with two lines the same length as
the section you removed, doing this to all three sides of the
triangle... and then doing it all again... and again... and again...
as you keep dividing and dividing the perimeter of the figure,
although the area of the interior is finite, the perimeter is
infinite!
An amazing list of links has been compiled by Chaffey High School, and
you can even listen to fractal music:
http://www.chaffey.org/fractals/
On the Quiddity Design Team's page, "What is a Fractal,"
http://www.dsoe.com/people/hoyle/fractal.html
they say: "Often fractals are self-similar, that is, they have the
property that each small portion of the fractal can be viewed as a
reduced-scale replica of the whole."
In the 1970s, Benoit Mandelbrot discovered fractal geometry and
adopted a more abstract definition of dimension than that used in
Euclidean geometry. He stated that when measuring the size of a
fractal, its dimension must be used as an exponent. Thus fractals are
not one- or two- or three- (or any other whole number-) dimensional,
but must be handled mathematically as if they have fractional
dimension.
Here are some more sites where you can to read about fractals and/or
find links to other Web pages about them:
Chopping Broccoli
http://www.glenbrook.k12.il.us/gbsmat/fractals/fractals.html
The sci.fractals FAQ
http://www.mta.ca/~mctaylor/sci.fractals-faq/
The Math Forum's page of fractal links
http://mathforum.org/library/browse/static/topic/fractals.html
Have fun surfing!
-Doctor Sarah, The Math Forum
Check out our web site! http://mathforum.org/dr.math/